The problem involves two functions, $f(x) = 2x^2$ and $g(x) = x + 4$, and asks for the following:

1. The expression for $(f + g)(x)$,
2. The expression for $(f \cdot g)(x)$,
3. The evaluation of $(f - g)(-2)$.

Step 1: Finding $(f + g)(x)$

By definition, $(f + g)(x) = f(x) + g(x)$.

Given:

• $f(x) = 2x^2$
• $g(x) = x + 4$

So: $(f + g)(x) = 2x^2 + (x + 4) = 2x^2 + x + 4$

Step 2: Finding $(f \cdot g)(x)$

By definition, $(f \cdot g)(x) = f(x) \cdot g(x)$.

So: $(f \cdot g)(x) = (2x^2) \cdot (x + 4) = 2x^2(x + 4) = 2x^3 + 8x^2$

Step 3: Evaluating $(f - g)(-2)$

By definition, $(f - g)(x) = f(x) - g(x)$.

So: $(f - g)(x) = 2x^2 - (x + 4) = 2x^2 - x - 4$ Now, substitute $x = -2$: $(f - g)(-2) = 2(-2)^2 - (-2 + 4) = 2(4) - (2) = 8 - 2 = 6$

Final Answers:

1. $(f + g)(x) = 2x^2 + x + 4$
2. $(f \cdot g)(x) = 2x^3 + 8x^2$
3. $(f - g)(-2) = 6$

Would you like further details on any of the steps?

Here are 5 related questions you might consider:

1. What is $(f - g)(x)$?
2. How do you evaluate $(f \cdot g)(-2)$?
3. What would the derivative of $(f + g)(x)$ be?
4. How does the function $(f \cdot g)(x)$ behave for large values of $x$?
5. What happens if you compute $(f / g)(x)$?

Tip: Always double-check your function evaluations by substituting carefully to avoid simple arithmetic mistakes.